# Rational curves on elliptic surfaces

I need a reference for the following assertion (if it's true). Let $X$ be a minimal elliptic surface over the field of complex numbers. Assume that its Kodaira dimension $\kappa(X)=1$. Then $X$ does not contain rational curves. Thanks!

• What if you had, say, a rational elliptic surface $Y$ with a reducible fibre containing a rational curve over a point $p_0$, then pulled back via a high-degree covering $C \rightarrow P^1$ that was unramified over $p_0$? – Bertie Mar 3 '17 at 17:00
• @Bertie - OOPS! I've forgotten to mention that the surface is minimal. – Yuri Zarhin Mar 3 '17 at 17:25
• Dear Zarhin: yes, I figured you meant minimal. But in fact I was talking about a reducible fibre which is a bunch of $-2$ curves.Anyway, abx seems to have settled the matter to your satisfaction. – Bertie Mar 4 '17 at 22:34
• Dear Bertie, thank you for your example, – Yuri Zarhin Mar 11 '17 at 3:59

I don't think this is true. Take a general pencil of cubics in $\mathbb{P}^2$, and blow-up the 9 fixed points to get an elliptic fibration $f:S\rightarrow \mathbb{P}^1$ which admits a section (at least 9 in fact). Pull back by a degree $n\geq 3$ covering $\mathbb{P}^1\rightarrow \mathbb{P}^1$ branched along two points $p_1,p_2$ such that $\ f^{-1}(p_i)\$ is smooth. You get a new fibration $f':S'\rightarrow \mathbb{P}^1$ with $\kappa (S')=1$, $S'$ minimal; the sections of $f$ pull back to sections of $f'$ which are smooth rational curves.